Why my Fully Connected Neural Network outputs the same prediction?
two hidden layers each comprising two neurons

From your description it looks like that you only have 6 parameters for your inner layer (2x2 weight matrix + 2 biases). The whole network should be easy to interpret: you've got two 13-dimensional weight vectors $\vec{w}_1,\vec{w}_2$ that are dot-multiplied with the inputs, plus two biases $b$ and activation $\sigma$:
$$ l_1 = \sigma\left(\vec{w}_1\vec{x} + b_1\right)$$
$$ l_2 = \sigma\left(\vec{w}_2\vec{x} + b_2\right)$$
Then these two values are multiplied by 2x2 matrix + biases, then activation and linear combination.
I'd look at how $l_1$ and $l_2$ are distributed. The fact that the outputs don't change is most likely due to the first layer getting saturated somehow. Look at the 13th dimension of $\vec{w}_i$ - it is likely to be large compared to other dimensions.
First thing I'd try - standardizing your input 13th dimension, so it is distributed closer to $[0,1]$ ( or $[-1,1]$ ) range.